Under "$HOME/.local/share/Steam/steamapps/common", locate the game folder of your interest.
Usually it has a file named steam_appid.txt, which stores the identifier used by steam to create the game save folder.
If steam_appid.txt is not present in the game folder, you can still identify the game id by searching it into "$HOME/.local/share/Steam/userdata/XXXXXXXXX" directory.
XXXXXXXXX is your steam ID. There you will find one folder for each game, the folder names are the game ids.
Inside each folder there is usually a remotecache.vdf file that has some informations on the game.
If you are lucky enough you will infer which game remotecache.vdf is referring to by reading its content.
Steams creates a new windows-like directory tree for each game save.
Games saves are stored inside "$HOME/.local/share/Steam/steamapps/compatdata" directory.
Use the $steam_appid to identify the correct game directory.
After that, the save files can be found in /pfx/drive_c/users/steamuser/AppData/Roaming/$GAME_NAME.
$GAME_NAME format varies game by game.
Just backup that directory and restore it when needed.